{"paper":{"title":"Retention Profiles and KL Contraction Bounds in Finite Markov Chains","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Saurav Jadhav","submitted_at":"2026-06-25T14:15:59Z","abstract_excerpt":"We study Kullback-Leibler (KL) contraction in finite Markov chains through a row-wise perspective. Evaluating the SDPI ratio at point masses yields a state-indexed retention profile $r(x)=D_{\\mathrm{KL}}(P(x,\\cdot)\\|\\pi)/\\log(1/\\pi(x))$ and a localization ratio $L(P)=\\bar r_\\pi/M\\in[0,1]$ (with $M=\\max_x r(x)$, $\\bar r_\\pi=\\mathbb{E}_\\pi r$) that distinguishes localized from global contraction obstructions. Our main contributions are (i) a convexity-gap identity showing that the gap between the row-averaged divergence and $D_{\\mathrm{KL}}(\\mu P\\|\\pi)$ equals the mutual information $I_\\mu(X;Y)$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.27073","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.27073/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}